Some properties of the nilradical and non-nilradical graphs over finite commutative ring Zn
Let Zn be the finite commutative ring of residue classes modulo n with identity and Γ(Zn) be its zero-divisor graph. In this paper, we investigated some properties of nilradical graph, denoted by N(Zn) and non-nilradical graph, denoted by Ω(Zn) of Γ(Zn). In particular, we determined the Chromatic nu...
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Інститут прикладної математики і механіки НАН України
2017
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| Цитувати: | Some properties of the nilradical and non-nilradical graphs over finite commutative ring Zn / S. Chandra, O. Prakash, S. Suthar // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 2. — С. 181-190. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1566412019-06-19T01:28:11Z Some properties of the nilradical and non-nilradical graphs over finite commutative ring Zn Chandra, S. Prakash, O. Suthar, S. Let Zn be the finite commutative ring of residue classes modulo n with identity and Γ(Zn) be its zero-divisor graph. In this paper, we investigated some properties of nilradical graph, denoted by N(Zn) and non-nilradical graph, denoted by Ω(Zn) of Γ(Zn). In particular, we determined the Chromatic number and Energy of N(Zn) and Ω(Zn) for a positive integer n. In addition, we have found the conditions in which N(Zn) and Ω(Zn) graphs are planar. We have also given MATLAB coding of our calculations. 2017 Article Some properties of the nilradical and non-nilradical graphs over finite commutative ring Zn / S. Chandra, O. Prakash, S. Suthar // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 2. — С. 181-190. — Бібліогр.: 8 назв. — англ. 1726-3255 2010 MSC:13Axx, 05Cxx, 05C15, 05C10, 65KXX. http://dspace.nbuv.gov.ua/handle/123456789/156641 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let Zn be the finite commutative ring of residue classes modulo n with identity and Γ(Zn) be its zero-divisor graph. In this paper, we investigated some properties of nilradical graph, denoted by N(Zn) and non-nilradical graph, denoted by Ω(Zn) of Γ(Zn). In particular, we determined the Chromatic number and Energy of N(Zn) and Ω(Zn) for a positive integer n. In addition, we have found the conditions in which N(Zn) and Ω(Zn) graphs are planar. We have also given MATLAB coding of our calculations. |
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Article |
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Chandra, S. Prakash, O. Suthar, S. |
| spellingShingle |
Chandra, S. Prakash, O. Suthar, S. Some properties of the nilradical and non-nilradical graphs over finite commutative ring Zn Algebra and Discrete Mathematics |
| author_facet |
Chandra, S. Prakash, O. Suthar, S. |
| author_sort |
Chandra, S. |
| title |
Some properties of the nilradical and non-nilradical graphs over finite commutative ring Zn |
| title_short |
Some properties of the nilradical and non-nilradical graphs over finite commutative ring Zn |
| title_full |
Some properties of the nilradical and non-nilradical graphs over finite commutative ring Zn |
| title_fullStr |
Some properties of the nilradical and non-nilradical graphs over finite commutative ring Zn |
| title_full_unstemmed |
Some properties of the nilradical and non-nilradical graphs over finite commutative ring Zn |
| title_sort |
some properties of the nilradical and non-nilradical graphs over finite commutative ring zn |
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Інститут прикладної математики і механіки НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/156641 |
| citation_txt |
Some properties of the nilradical and non-nilradical graphs over finite commutative ring Zn / S. Chandra, O. Prakash, S. Suthar // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 2. — С. 181-190. — Бібліогр.: 8 назв. — англ. |
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Algebra and Discrete Mathematics |
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2025-07-14T09:01:34Z |
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1837612342849830912 |
| fulltext |
“adm-n4” 11:26 page #3
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 24 (2017). Number 2, pp. 181–190
© Journal “Algebra and Discrete Mathematics”
Some properties
of the nilradical and non-nilradical graphs
over finite commutative ring Zn
Shalini Chandra, Om Prakash and Sheela Suthar
Communicated by M. Ya. Komarnytskyj
Abstract. Let Zn be the finite commutative ring of residue
classes modulo n with identity and Γ(Zn) be its zero-divisor graph.
In this paper, we investigate some properties of nilradical graph,
denoted by N(Zn) and non-nilradical graph, denoted by Ω(Zn) of
Γ(Zn). In particular, we determine the Chromatic number and
Energy of N(Zn) and Ω(Zn) for a positive integer n. In addition,
we have found the conditions in which N(Zn) and Ω(Zn) graphs are
planar. We have also given MATLAB coding of our calculations.
Introduction
The concept of zero-divisor graph was introduced by I. beck in 1988 but
the most common definition of zero-divisor graph given by D. F. Anderson
and P. S. Livingston in 1999 is as follows: “Let R be a commutative ring
(with 1) and let Z(R) be its set of zero-divisors. We associate a simple
graph Γ(R) to R with vertices Z(R)∗ = Z(R) − {0}, the set of nonzero
zero-divisors of R, and for distinct x, y ∈ Z(R)∗, the vertices x and y are
adjacent if and only if xy = 0. Thus, Γ(R) is the empty graph if and only
if R is an integral domain.” We have derived some results for the ring Zn.
A complete graph is a graph (without loops and multiple edges) in
which every vertex is adjacent to any other vertices of the graph. A graph
in which all vertices have the same degree is said to be a regular graph.
A complete bipartite graph is a graph whose vertices can be divided into
2010 MSC: 13Axx, 05Cxx, 05C15, 05C10, 65KXX.
Key words and phrases: commutative ring, zero-divisor graph, nilradical graph,
non-nilradical graph, chromatic number, planar graph, energy of a graph.
“adm-n4” 11:26 page #4
182 Nilradical and Non-nilradical Graphs over Zn
two sets such that every vertex in one set is connected to every vertex in
the other, and no vertex is connected to any other vertices in the same set.
A star graph is a complete bipartite graph in which at least one of the two
vertex sets contains only one vertex. That one vertex is called the center of
the star graph. A vertex of a graph is isolated if there is no edge incident
on it. A graph is almost connected if there exists a path between any two
non-isolated vertices. A proper coloring of a graph Zn is a function that
assigns a color to each vertex such that no any two adjacent vertices have
the same color. The chromatic number of Zn, denoted by χ(Zn), is the
smallest number of colors required for proper coloring. A planar graph is
a graph that can be embedded in the plane, i.e, it can be drawn on the
plane in such a way that its edges intersect only at their endpoints and
we will repeatedly use Kuratowski’s theorem, which states that a graph is
planar if and only if it does not contain a subdivision of K5 or K3,3. The
energy of a graph is the sum of absolute value of all eigenvalues of the
adjacency matrix. The adjacency matrix corresponding to a zero divisor
graph is defined as A = [ai.j ], where ai,j = 1, if vi & vj represent zero
divisor, i.e., vi.vj = 0 and ai,j = 0 otherwise, where vi and vj are vertices
of the graph.
Nilradical and non-nilradical graphs
Definition 1.1. The nilradical graph of Zn, denoted by N(Zn), is the
graph whose vertices are the nonzero nilpotent elements of Zn and any
two vertices are connected by an edge if and only if their product is 0.
Definition 1.2. The non-nilradical graph of Zn, denoted by Ω(Zn), is
the graph whose vertices are the non-nilpotent zero-divisors of Zn and
any two vertices are connected by an edge if and only if their product
is 0.
1. Chromatic number and planarity of nilradical
and non-nilradical graphs
Theorem 1. If p and q are distinct prime numbers and n is a positive
integer, then
(1) χ(N(Zn)) = 0 if n = pq;
(2) χ(N(Zn)) = p − 1 if n = p2;
(3) χ(N(Zn)) = pq − 1 if n = p2q2;
(4) χ(N(Zn)) = p if n = p3;
(5) χ(N(Zn)) = p − 1 if n = p2q.
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S. Chandra, O. Prakash, S. Suthar 183
Proof. (1) Let n = pq, where p and q are distinct primes. Then N(Zn) is
an empty graph. So, there is no need of any color for coloring the graph.
Hence, chromatic number is zero.
(2) Let n = p2, where p is a prime number. If p = 2, then N(Zn) has
only one vertex. This implies the chromatic number is one. If p > 3, then
the number of nilpotent elements which are divisible by p2 are (p − 1).
Also, these (p − 1) nilpotent elements form a complete graph. So, (p − 1)
colors are required for coloring the graph and these (p − 1) colors are
minimum in numbers. Therefore, chromatic number is (p − 1).
(3) Let n = p2q2, where p and q are prime numbers and p 6= q. Then
the nilpotent elements are multiple of pq and number of nilpotent elements
are pq − 1. Also, these pq − 1 elements are connected to each other. Thus,
pq − 1 elements form a complete graph with pq − 1 vertices. Therefore,
(pq − 1) colors are required for coloring the graph. Hence, chromatic
number of N(Zp2q2) is (pq − 1).
(4) If n = p3, where p is a prime number, then N(Zn) is a complete
p-partite graph with (p2 − 1) vertices. Therefore, we required p colors for
proper coloring. Hence, chromatic number of N(Zn) is p.
(5) Let n = p2q, where p and q are distinct prime numbers. Then
the nilpotent elements are multiple of pq, and the number of nilpotent
elements are (p − 1). These (p − 1) elements are connected to each other
and form a complete graph with (p − 1) vertices. Therefore, (p − 1) colors
are required for coloring the graph N(Zp2q). Hence, chromatic number of
N(Zp2q) is (p − 1).
Theorem 2. Let p and q be two distinct prime numbers and n a positive
integer. Then
(1) χ(Ω(Zn)) = m if n = p1p2p3 . . . pm, m > 1, where p1, p2, . . . , pm are
distinct primes;
(2) χ(Ω(Zn)) = 0 if n = p2;
(3) χ(Ω(Zn)) = 0 if n = p3;
(4) χ(Ω(Zn)) = 2 if n = p2q, for q = 2 or 3.
Proof. (1) Let n = p1 p2 p3 . . . pm, for some positive integer m, such that
all pi are distinct prime numbers. Then Ω(Zn) is equal to Γ(Zn) and since
Γ(Zn) is m-partite graph, therefore Ω(Zn) is also m-partite graph. In this
case, m distinct colors are needed for proper coloring of the graph Ω(Zn).
Thus, Chromatic number of graph Ω(Zn) is m.
(2) Let n = p2, where p is a prime number. Then clearly Ω(Zn) is an
empty graph. Hence, there is no need of any color for coloring the graph
Ω(Zn). Hence, chromatic number is zero.
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184 Nilradical and Non-nilradical Graphs over Zn
(3) Let n = p3, where p is a prime number. Then Ω(Zn) is an empty
graph. Hence, there is no need of any color for coloring the graph Ω(Zn).
So, chromatic number is zero.
(4) Let n = p2q, where p and q are distinct prime numbers. Then
multiple of p, p2 and q2 are not adjacent to themselves. But the vertices
which are multiple of p2 are adjacent to those vertices which are multiple
of q and not adjacent with multiple of p. Similarly, elements which are
multiple of q are not adjacent with multiple of p. Thus, there are two
disjoint sets of vertices which are adjacent from one set to other but
not adjacent to each other in a set. Therefore, two colors are required
for coloring the Ω(Zn) graph and also we can use one color from them
for isolated vertices. Hence, chromatic number is two for Ω(Zn), when
n = p2q, where p, q are distinct prime numbers.
Theorem 3. If p and q are distinct prime numbers and n is a positive
integer, then
(1) N(Zn) is planar, where n = pq;
(2) N(Zn) is planar for p 6 5 and non-planar for p > 5, where n = p2;
(3) N(Zn) is planar for p 6 5 and q is any prime number, where
n = p2q;
(4) N(Zn) is planar, if p < 5 and non-planar for p > 5, where n = p3;
(5) N(Zn) is planar, where n = 4k, gcd(2, k) = 1, p2 6 | k for any prime
p and k is any positive integer;
(6) N(Zn) is planar, where n = 9k, gcd(3, k) = 1, p2 6 | k for any prime
p and k is any positive integer.
Proof. (1) If n = pq, where p and q are distinct prime numbers, then
N(Zn) is an empty graph. Therefore, N(Zn) graph is a planar graph.
(2) If n = p2, where p is a prime number, then the nilpotent elements
of (Zn) are multiple of p. So, there are (p − 1) nilpotent elements which
form a complete graph with (p − 1) vertices and all vertices are adjacent
to each other. If p = 2, then N(Zn) has only one vertex and when p = 3,
then N(Zn) has two vertices. In this case, N(Zn) is a planar graph. If
p = 5, then N(Zn) is a complete graph with 4 vertices and all vertices
are adjacent to each other. Therefore, N(Zn) is a planar graph.
For p > 5, N(Zn) graph contains K3,3 or K5 as a proper subgraph.
Hence, N(Zn) is not a planar graph for p > 5.
(3) If n = p2q, where p and q are distinct prime numbers, then N(Zn)
is a complete graph with (p − 1) vertices. Thus, N(Zn) is a planar graph
only when p 6 5 and q is any prime, p 6= q, otherwise N(Zn) contains
“adm-n4” 11:26 page #7
S. Chandra, O. Prakash, S. Suthar 185
K5 as a subgraph which is not planar and therefore N(Zn) is a planar if
p 6 5.
(4) If n = p3, where p is any prime, then N(Zn) is a complete p-partite
graph with (p2 − 1) vertices. Therefore, N(Zn) is planar for p < 5 and
non-planar for p > 5.
(5) If n = 4k, and p2 6 | k, for a prime p and k is any positive integer,
then N(Zn) has only one vertex, hence N(Zn) graph is a planar graph.
(6) If n= 9k, p2 6 | k, for all prime p and k is any positive integer, then
N(Zn) has two vertices which are adjacent to each other. Thus, N(Zn) is
a planar graph.
Theorem 4. If p and q are distinct prime numbers and n is a positive
integer, then
(1) Ω(Zn) is not planar, for n = pq, (specially p > 5 and q > 3);
(2) Ω(Zn) is planar, for n = p2;
(3) Ω(Zn) is planar, for n = p3;
(4) Ω(Zn) is planar for k 6 6 and non-planar for all k > 6, where
n = 4k, gcd(2, k) = 1 and p2 6 | k, for a prime p and k is any
positive integer;
(5) Ω(Zn) is a planar for k 6 4 and non-planar for k > 5, where n = 9k,
gcd(3, k) = 1 and p2 6 | k, for a prime p and k is any positive integer;
(6) Ω(Zn) is planar for q = 2 and 3, and p is any prime number, where
n = p2q.
Proof. (1) Let n = pq, such that p and q are distinct primes. Then clearly
Ω(Zn) is a bi-partite graph. If, we take n = pq where p = 2 and q is any
prime number, then Ω(Zn) is a star graph. We know that star graph is a
planar graph. Hence, Ω(Zn) is a planar graph in this case. If p = 3 and q
is any prime number, then Ω(Zn) is a complete bi-partite graph, which
is a planar graph. If p > 5 and q is any prime number which is greater
than 3, then Ω(Zn) is not a planar graph. Because, in this case, Ω(Zn)
graph contain K3,3 as a subgraph. Therefore, Ω(Zn) is not a planar graph
for n = pq.
(2) Let n = p2, where p is any prime number. Then, there are no
non-nilpotent elements of Zn in Ω(Zn). Therefore, Ω(Zn) is an empty
graph. Hence, Ω(Zn) is a planar graph.
(3) Let n = p3, where p is any prime number. Then, there is no non-
nilpotent element of Zn in Ω(Zn). Therefore, Ω(Zn) is an empty graph.
Hence, Ω(Zn) is a planar graph.
(4) Let n = 4k, where p2 6 | k, for a prime p and k is any positive
integer. Then, Ω(Zn) is planar for k 6 6. If we take k is any prime number,
“adm-n4” 11:26 page #8
186 Nilradical and Non-nilradical Graphs over Zn
then Ω(Zn) is always complete bi-partite graph. We know that complete
bi-partite graph is planar graph. Therefore, Ω(Zn) is the planar graph
for the prime k. On the other hand, if k > 6, then Ω(Zn) graph contains
K3,3 or K5 as a subgraph. Thus, for k > 6, Ω(Zn) graph is not a planar.
(5) Let n = 9k, where p2 6 | k, for a prime p and k is any positive
integer. Then Ω(Zn) is a planar graph for k 6 4. For k > 5, Ω(Zn) graph
contains K3,3 as a subgraph. Therefore, graph is not a planar for k > 5.
(6) Let n = p2q, where p and q are distinct primes. If q = 2 and p is
any prime number, then Ω(Zn) graph is a star graph. Therefore, Ω(Zn)
graph is planar. If q = 3 and p is any prime number, then Ω(Zn) graph is
a complete bi-partite graph. Therefore, Ω(Zn) is planar graph. For q > 5
and p is any prime greater than 2 (and 3), Ω(Zn) graph contains K3,3 or
K5 as a subgraph. Thus, Ω(Zn) is non-planar.
Lemma 1. If n = pq, where p and q are primes, then there is no isolated
vertex in Ω(Zn) graph.
Proof. If n = pq, where p and q are distinct primes, then Ω(Zn) is a
complete bi-partite graph. Hence, there is no isolated vertex. When
n = p2, for any prime p, then there is no vertex in Ω(Zn). Hence, graph
is empty. Thus, in this case again we have no isolated vertex.
Lemma 2. If n = p3, for any prime p, then Ω(Zn) graph has no isolated
vertex.
Proof. If n = p3, then zero divisor graph has p2 − 1 elements in which all
elements are nilpotent and no element is non-nilpotent. Also all nilpotent
elements are adjacent with nilpotent elements, but in Ω(Zn), there are
no non-nilpotent elements. Thus, Ω(Zn) is an empty graph. Therefore,
Ω(Zn) graph has no isolated vertex.
Observation 1. If n = p2q, for p and q are distinct prime numbers, then
Ω(Zn) graph has (p − 1)(q − 1) isolated vertices.
2. Energy of nilradical and non-nilradical graphs
Theorem 5. If n = p2, for prime p, then E(N(Zn)) is (2p − 4) and
E(Ω(Zn)) is zero (E(Ω(Zn)) is zero also for p3).
Proof. When n = p2, N(Zn) is a complete graph with p−1 vertices. Then
f(λ) =| λIp−1 − M(N(Zn)) |= (λ − 1)p−2(λ + p − 2) by [2], where M
“adm-n4” 11:26 page #9
S. Chandra, O. Prakash, S. Suthar 187
is a matrix of order (p − 1). If f(λ) = 0, then λ = 1, 2 − p. Therefore,
Σp−1
i=1
| λi |= 2p − 4.
When n = p2, then Ω(Zn) graph is an empty graph. Hence, it has
zero energy.
When n = p3, then Ω(Zn) is an empty graph and hence, it has zero
energy.
Theorem 6. If n = pq, where p and q are distinct primes, then energy
of Ω(Zn) is 2
√
(p − 1)(q − 1) and energy of N(Zn) is zero.
Proof. Let n = pq, where p and q are two distinct prime. Then Ω(Zn)
is a bi-partite graph. Also, its eigen polynomial f(λ) =| λIp+q−2 -
M(Ω(Zn)) |= (λ)p+q−4(λ2 − (p − 1)(q − 1)), where M is a matrix of
order (p + q − 2). Thus, nonzero eigenvalues are ±
√
(p − 1)(q − 1) and
so E(Ω(Zn)) = 2
√
(p − 1)(q − 1). Also, N(Zn)) graph has no vertices for
distinct primes p and q. Thus, E(N(Zn)) has no energy.
Theorem 7. For n = p2q, energy of N(Zn) is 2p − 4, for all distinct
primes p and q.
Proof. Same as above Theorem (5).
Observation 2. If n = p2q, then energy of Ω(Zn) is:
(1) 2
√
pq − 2, for p = 2 and q is any prime number;
(2) 2
√
pq + p(q − 2), for p = 3 and q is any prime number;
(3) 2
√
2pq + 2p(q − 2), for p = 5 and q is any prime number.
3. Computer program
Now, we offer three algorithms for calculating energy with MATLAB
software. These algorithms include several sub-algorithms. It is enough
to input n. In the first algorithm at the first stage, we obtain M(N(Zn))
and plot N(Zn) by function nil_radical_zn2(p). At the second stage,
we calculate Energy index by using energy.
In the second algorithm at the first stage, we obtain Ω(N(Zn)) and
plot Ω(Zn) by function non_nil_radical_zn2(p). At the second stage,
we calculate Energy index by using energy.
In third algorithm, we put the value of n and call above two functions
together.
“adm-n4” 11:26 page #10
188 Nilradical and Non-nilradical Graphs over Zn
First algorithm
function Nz=ni l_rad ica l_zn2 (p)
n=p ;
M= [ ] ;
for i =1:n−1
for j =1:n−1
i f mod( i ∗ i , n)==0
M=[M, i ] ;
break ;
end
end
end
M
n=length (M) ;
for i =0:n−1
axes ( i +1 ,:)=[ cos (2∗ pi∗ i /n ) , sin (2∗ pi∗ i /n ) ] ;
end
Nz=zeros (n ) ;
hold on
for i =1:n
plot ( axes ( i , 1 ) , axes ( i , 2 ) , ’ ∗ ’ )
i f mod(M( i )^2 ,p)==0
Nz( i , i )=1;
plot ( axes ( i , 1 ) , axes ( i , 2 ) , ’ ro ’ )
end
end
for i =1:n−1
for j=i +1:n
i f mod(M( i )∗M( j ) , p)==0
Nz( i , j )=1; Nz( j , i )=1;
plot ( axes ( [ i , j ] , 1 ) , axes ( [ i , j ] , 2 ) ) ;
end
end
end
eg=eig (Nz)
E=sum( abs ( eg ) )
Second algorithm
function NNz=non_nil_radical_zn2 (p)
n=p ;
M= [ ] ;
for i =1:n−1
for j =1:n−1
i f mod( i ∗ j , n)==0
“adm-n4” 11:26 page #11
S. Chandra, O. Prakash, S. Suthar 189
i f mod( i ∗ i , n)~=0
M=[M, i ] ;
break ;
end
end
end
end
M
n=length (M) ;
for i =0:n−1
axes ( i +1 ,:)=[ cos (2∗ pi∗ i /n ) , sin (2∗ pi∗ i /n ) ] ;
end
NNz=zeros (n ) ;
hold on
for i =1:n
plot ( axes ( i , 1 ) , axes ( i , 2 ) , ’ ∗ ’ )
i f mod(M( i )^2 ,p)==0
NNz( i , i )=1;
plot ( axes ( i , 1 ) , axes ( i , 2 ) , ’ ro ’ )
end
end
for i =1:n−1
for j=i +1:n
i f mod(M( i )∗M( j ) , p)==0
NNz( i , j )=1; NNz( j , i )=1;
plot ( axes ( [ i , j ] , 1 ) , axes ( [ i , j ] , 2 ) ) ;
end
end
end
eg=eig (NNz)
E=sum( abs ( eg ) )
Third algorithm
p=n ;
Nz=ni l_rad ica l_zn2 (p)
f igure ;
NNz=non_nil_radical_zn2 (p)
f igure ;
All above algorithms are also useful for p3. If we use the formula
“if mod(i*j,n)==0” at the place of sixth line in the first algorithm, then
it will give fruitful result for p3.
“adm-n4” 11:26 page #12
190 Nilradical and Non-nilradical Graphs over Zn
n E(N(Zn)) E(Ω(Zn))
27 7.2111 0
45 2 9.7980
77 0 15.4919
121 18 0
225 26.00 21.9089
343 32.3110 0
Table 1. The values of E(N(Zn)) and E(Ω(Zn)) for n = 27, 45, 77, 121, 225
and 343.
References
[1] M. R. Ahmadi and R. J. Nezhad, Energy and Wiener Index of Zero Divisor Graphs,
Iranian J. Math. Chem., 2(1), 2011, pp.45-51.
[2] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring,
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Contact information
Shalini Chandra,
Sheela Suthar
Department of Mathematics and Statistics,
Banasthali Vidyapith,
Banasthali, Rajasthan - 304 022, India
E-Mail(s): chandrshalini@gmail.com,
sheelasuthar@gmail.com
Web-page(s): www.banasthali.ac.in
Om Prakash Department of Mathematics, IIT Patna,
Patliputra colony, Patna - 800 013, India
E-Mail(s): om@iitp.ac.in
Web-page(s): www.iitp.ac.in
Received by the editors: 24.09.2015
and in final form 25.02.2016.
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